Answer: (a) To find the scale factor from the map to the actual trail, we can use the information that 1 cm on the first map represents 2 km on the actual trail. The scale factor is the ratio of the length on the map to the length on the actual trail.
Scale factor = Length on map / Length on actual trail
For the first map:
Length on map = 8 cm
Length on actual trail = 8 cm * 2 km/cm = 16 km
Scale factor = 8 cm / 16 km = 1/2,000,000
So, the scale factor from the first map to the actual trail is 1/2,000,000. The length of the actual trail is 16 km.
(b) To find the scale factor from the first map to the second map, we need to compare the lengths of the corresponding sides on both maps.
For the first map:
Side lengths of the landmark = 3 mm, 4 mm, and 5 mm
For the second map:
Let the side lengths of the corresponding landmark on the second map be x mm, y mm, and z mm.
The scale factor from the first map to the second map is the ratio of the lengths of the corresponding sides.
Scale factor = Length on second map / Length on first map
For the side with a length of 3 mm:
Scale factor = x mm / 3 mm
For the side with a length of 4 mm:
Scale factor = y mm / 4 mm
For the side with a length of 5 mm:
Scale factor = z mm / 5 mm
Since the landmark is a triangle, the side lengths are related by the Pythagorean theorem:
(3 mm)^2 + (4 mm)^2 = (5 mm)^2
9 + 16 = 25
So, the landmark on the second map is also a triangle with side lengths of 3 mm, 4 mm, and 5 mm. Therefore, the scale factor from the first map to the second map is 1:1, and the side lengths of the landmark on the second map are also 3 mm, 4 mm, and 5 mm.